 reserve X,Y,Z,E,F,G,S,T for RealLinearSpace;
 reserve X,Y,Z,E,F,G for RealNormSpace;
 reserve S,T for RealNormSpace-Sequence;

theorem IS01A:
  for u be Lipschitzian MultilinearOperator of <*E,F*>,G holds
  u * (IsoCPNrSP(E,F)) is Lipschitzian BilinearOperator of E,F,G
  proof
    let u be Lipschitzian MultilinearOperator of <*E,F*>,G;
    reconsider L = u * (IsoCPNrSP(E,F)) as Function of [:E,F:],G;
    dom <*E,F*> = Seg len <*E,F*> by FINSEQ_1:def 3
    .= {1,2} by FINSEQ_1:2,44; then
reconsider i1 = 1, i2 = 2 as Element of dom <*E,F*> by TARSKI:def 2;
    A2: for x1,x2 be Point of E, y be Point of F
        holds L.(x1+x2,y) = L.(x1,y) + L.(x2,y)
    proof
      let x1,x2 be Point of E, y be Point of F;
      reconsider xy = <*x1,y*> as Point of product <*E,F*> by PRVECT_3:19;
      reconsider L1 = u * reproj(i1,xy)
        as LinearOperator of E,G by LOPBAN10:def 6;
      A5: dom reproj(i1,xy) = the carrier of <*E,F*>.i1 by FUNCT_2:def 1
       .= the carrier of E;
      len xy = 2 by FINSEQ_1:44; then
      dom xy = {1,2} by FINSEQ_1:2,def 3; then
      A7: i1 in dom xy by TARSKI:def 2;
      A8: len(xy +* (i1,x1+x2)) = 2 by LemmaA;
      A9: (xy +* (i1,x1+x2)).1 = x1+x2 by A7,FUNCT_7:31;
      A10: (xy +* (i1,x1+x2)).2 = xy.i2 by FUNCT_7:32
      .= y;
      reconsider x1x2 = x1+x2 as Element of <*E,F*>.i1;
      A12: reproj(i1,xy).x1x2 = xy +* (i1,x1+x2) by NDIFF_5:def 4
      .= <*x1+x2, y*> by A8,A9,A10,FINSEQ_1:44;
      A13: len(xy +* (i1,x1)) = 2 by LemmaA;
      A14: (xy +* (i1,x1)).1 = x1 by A7,FUNCT_7:31;
      A15: (xy +* (i1,x1)).2 = xy.i2 by FUNCT_7:32
      .= y;
      A17:reproj(i1,xy).x1 = xy +* (i1,x1) by NDIFF_5:def 4
      .= <*x1,y*> by A13,A14,A15,FINSEQ_1:44;
      A18: len(xy +* (i1,x2)) = 2 by LemmaA;
      A19: (xy +* (i1,x2)).1 = x2 by A7,FUNCT_7:31;
      A20: (xy +* (i1,x2)).2 = xy.i2 by FUNCT_7:32
      .= y;
      A22: reproj(i1,xy).x2 = xy +* (i1,x2) by NDIFF_5:def 4
      .= <*x2,y*> by A18,A19,A20,FINSEQ_1:44;
      A4: [x1+x2,y] is Point of [:E,F:]
         & [x1,y] is Point of [:E,F:] & [x2,y] is Point of [:E,F:]; then
      A23: L.(x1+x2,y) = u.((IsoCPNrSP(E,F)).(x1+x2,y)) by FUNCT_2:15
      .= u.(reproj(i1,xy).(x1+x2)) by A12,NDIFF_7:def 3
      .= L1.(x1+x2) by A5,FUNCT_1:13;
      A24: L.(x1,y) = u.((IsoCPNrSP(E,F)).(x1,y)) by A4,FUNCT_2:15
      .= u.(reproj(i1,xy).x1) by A17,NDIFF_7:def 3
      .= L1.x1 by A5,FUNCT_1:13;
      L.(x2,y) = u.((IsoCPNrSP(E,F)).(x2,y)) by A4,FUNCT_2:15
      .= u.(reproj(i1,xy).x2) by A22,NDIFF_7:def 3
      .= L1.x2 by A5,FUNCT_1:13;
      hence L.(x1+x2,y) = L.(x1,y) + L.(x2,y) by A23,A24,VECTSP_1:def 20;
    end;
    A26: for x be Point of E, y be Point of F, a be Real
        holds L.(a*x, y) = a * L.(x,y)
    proof
      let x be Point of E, y be Point of F, a be Real;
      reconsider xy = <*x,y*> as Point of product <*E,F*> by PRVECT_3:19;
      reconsider L1 = u * reproj(i1,xy)
        as LinearOperator of E,G by LOPBAN10:def 6;
      A29: dom reproj(i1,xy) = the carrier of <*E,F*>.i1 by FUNCT_2:def 1
      .= the carrier of E;
      len xy = 2 by FINSEQ_1:44; then
      dom xy = {1,2} by FINSEQ_1:2,def 3; then
      A31: i1 in dom xy by TARSKI:def 2;
      A32: len(xy +* (i1,a*x)) = 2 by LemmaA;
      A33: (xy +* (i1,a*x)).1 = a*x by A31,FUNCT_7:31;
      A34: (xy +* (i1,a*x)).2 = xy.i2 by FUNCT_7:32
      .= y;
      reconsider x1x2 = a*x as Element of <*E,F*>.i1;
      A36: reproj(i1,xy).x1x2 = xy +* (i1,a*x) by NDIFF_5:def 4
      .= <*a*x,y*> by A32,A33,A34,FINSEQ_1:44;
      A37: len (xy +* (i1,x)) = 2 by LemmaA;
      A38: (xy +* (i1,x)).1 = x by A31,FUNCT_7:31;
      A39: (xy +* (i1,x)).2 = xy.i2 by FUNCT_7:32
      .= y;
      A41: reproj(i1,xy).x = xy +* (i1,x) by NDIFF_5:def 4
      .= <*x,y*> by A37,A38,A39,FINSEQ_1:44;
      A28: [x,y] is Point of [:E,F:] & [a*x,y] is Point of [:E,F:]; then
      A42: L.(a*x,y) = u.((IsoCPNrSP(E,F)).(a*x,y)) by FUNCT_2:15
      .= u.(reproj(i1,xy).(a*x)) by A36,NDIFF_7:def 3
      .= L1.(a*x) by A29,FUNCT_1:13;
      L.(x,y) = u.((IsoCPNrSP(E,F)).(x,y)) by A28,FUNCT_2:15
      .= u.(reproj(i1,xy).x) by A41,NDIFF_7:def 3
      .= L1.x by A29,FUNCT_1:13;
      hence L.(a*x, y) = a * L.(x,y) by A42,LOPBAN_1:def 5;
    end;
    A44: for x be Point of E, y1,y2 be Point of F
        holds L.(x,y1+y2) = L.(x,y1) + L.(x,y2)
    proof
      let x be Point of E, y1,y2 be Point of F;
      reconsider xy = <*x,y1*> as Point of product <*E,F*> by PRVECT_3:19;
      reconsider L1 = u * reproj(i2,xy)
        as LinearOperator of F,G by LOPBAN10:def 6;
      A47: dom reproj(i2,xy) = the carrier of <*E,F*>.i2 by FUNCT_2:def 1
      .= the carrier of F;
      len xy = 2 by FINSEQ_1:44; then
      dom xy = {1,2} by FINSEQ_1:2,def 3; then
      A49: i2 in dom xy by TARSKI:def 2;
      A50: len (xy +* (i2,y1+y2)) = 2 by LemmaA;
      A51: (xy +* (i2,y1+y2)).1 = xy.i1 by FUNCT_7:32
      .= x;
      A52: (xy +* (i2,y1+y2)).2 = y1+y2 by A49,FUNCT_7:31;
      reconsider x1x2 = y1+y2 as Element of <*E,F*>.i2;
      A54: reproj(i2,xy).x1x2 = xy +* (i2,y1+y2) by NDIFF_5:def 4
      .= <*x,y1+y2*> by A50,A51,A52,FINSEQ_1:44;
      A55: len(xy +* (i2,y1)) = 2 by LemmaA;
      A56: (xy +* (i2,y1)).1 = xy.i1 by FUNCT_7:32
      .= x;
      A57: (xy +* (i2,y1)).2 = y1 by A49,FUNCT_7:31;
      A59:reproj(i2,xy).y1 = xy +* (i2,y1) by NDIFF_5:def 4
      .= <*x,y1*> by A55,A56,A57,FINSEQ_1:44;
      A60: len (xy +* (i2,y2)) = 2 by LemmaA;
      A61: (xy +* (i2,y2)).1 = xy.i1 by FUNCT_7:32
      .= x;
      A62: (xy +* (i2,y2)).2 = y2 by A49,FUNCT_7:31;
      A64: reproj(i2,xy).y2 = xy +* (i2,y2) by NDIFF_5:def 4
      .= <*x,y2*> by A60,A61,A62,FINSEQ_1:44;
      A46: [x,y1+y2] is Point of [:E,F:] & [x,y1] is Point of [:E,F:]
         & [x,y2] is Point of [:E,F:]; then
      A65: L.(x,y1+y2) = u.((IsoCPNrSP (E,F)).(x,y1+y2)) by FUNCT_2:15
      .= u.(reproj(i2,xy).(y1+y2)) by A54,NDIFF_7:def 3
      .= L1.(y1+y2) by A47,FUNCT_1:13;
      A66: L.(x,y1) = u.((IsoCPNrSP (E,F)).(x,y1)) by A46,FUNCT_2:15
      .= u.(reproj(i2,xy).y1) by A59,NDIFF_7:def 3
      .= L1.y1 by A47,FUNCT_1:13;
      L.(x,y2) = u.((IsoCPNrSP (E,F)).(x,y2)) by A46,FUNCT_2:15
      .= u.(reproj(i2,xy).y2) by A64,NDIFF_7:def 3
      .= L1.y2 by A47,FUNCT_1:13;
      hence L.(x,y1+y2) = L.(x,y1) + L.(x,y2) by A65,A66,VECTSP_1:def 20;
    end;
    for x be Point of E, y be Point of F, a be Real
      holds L.(x, a*y) = a * L.(x,y)
    proof
      let x be Point of E, y be Point of F, a be Real;
      reconsider xy = <*x,y*> as Point of product <*E,F*> by PRVECT_3:19;
      reconsider L1 = u * reproj(i2,xy)
        as LinearOperator of F,G by LOPBAN10:def 6;
      A70: dom reproj(i2,xy) = the carrier of <*E,F*>.i2 by FUNCT_2:def 1
      .= the carrier of F;
      len xy = 2 by FINSEQ_1:44; then
      dom xy = {1,2} by FINSEQ_1:2,def 3; then
      A72: i2 in dom xy by TARSKI:def 2; then
      A75: (xy +* (i2, a*y)).2 = a * y by FUNCT_7:31;
      A73: len(xy +* (i2,a*y)) = 2 by LemmaA;
      A74: (xy +* (i2,a*y)).1 = xy.i1 by FUNCT_7:32
      .= x;
      reconsider x1x2 = a * y as Element of <*E,F*>.i2;
      A77:reproj(i2,xy).x1x2 = xy +* (i2,a*y) by NDIFF_5:def 4
      .= <*x, a*y*> by A73,A74,A75,FINSEQ_1:44;
      A78: len(xy +* (i2,y)) = 2 by LemmaA;
      A79: (xy +* (i2,y)).1 = xy.i1 by FUNCT_7:32
      .= x;
      A80: (xy +* (i2,y)).2 = y by A72,FUNCT_7:31;
      A82:reproj(i2,xy).y = xy +* (i2,y) by NDIFF_5:def 4
      .= <*x,y*> by A78,A79,A80,FINSEQ_1:44;
      A69: [x,y] is Point of [:E,F:] & [x,a*y] is Point of [:E,F:]; then
      A83: L.(x, a*y) = u.((IsoCPNrSP(E,F)).(x,a*y)) by FUNCT_2:15
      .= u.(reproj(i2,xy).(a*y)) by A77,NDIFF_7:def 3
      .= L1.(a*y) by A70,FUNCT_1:13;
      L.(x,y) = u.((IsoCPNrSP(E,F)).(x,y)) by A69,FUNCT_2:15
      .= u.(reproj(i2,xy).y) by A82,NDIFF_7:def 3
      .= L1.y by A70,FUNCT_1:13;
      hence L.(x,a*y) = a * L.(x,y) by A83,LOPBAN_1:def 5;
    end; then
    reconsider L as BilinearOperator of E,F,G by A2,A26,A44,LOPBAN_8:12;
    ex K being Real
    st 0 <= K
     & for x being VECTOR of E, y being VECTOR of F
       holds ||. L.(x,y) .|| <= K * ||. x .|| * ||. y .||
    proof
      consider K being Real such that
      A85: 0 <= K
       & for s being Point of product <*E,F*>
         holds ||. u.s .|| <= K * NrProduct s by LOPBAN10:def 10;
      take K;
      thus 0 <= K by A85;
      let x be VECTOR of E, y be VECTOR of F;
      reconsider xy = <*x,y*> as Point of product <*E,F*> by PRVECT_3:19;
      [x,y] is Point of [:E,F:]; then
      A87: L.(x,y) = u.((IsoCPNrSP(E,F)).(x,y)) by FUNCT_2:15
      .= u.(<*x,y*>) by NDIFF_7:def 3;
      reconsider s = <*x,y*> as Point of product <*E,F*> by PRVECT_3:19;
      consider Nx be FinSequence of REAL such that
      A88: dom Nx = dom <*E,F*>
         & (for i be Element of dom <*E,F*> holds Nx.i = ||.s.i.||)
         & NrProduct s = Product Nx by LOPBAN10:def 9;
      dom Nx = Seg len <*E,F*> by A88,FINSEQ_1:def 3
      .= Seg 2 by FINSEQ_1:44; then
      A89: len Nx = 2 by FINSEQ_1:def 3;
      A91: Nx.1 = ||.s.i1.|| by A88
      .= ||.x.||;
      Nx.2 = ||.s.i2.|| by A88
      .= ||.y.||; then
      Nx = <* ||.x.||, ||.y.|| *> by A89,A91,FINSEQ_1:44; then
      Product Nx = ||.x.|| * ||.y.|| by RVSUM_1:99; then
      ||.L.(x,y).|| <= K * (||.x.|| * ||.y.||) by A85,A87,A88;
      hence thesis;
    end;
    hence thesis by LOPBAN_9:def 3;
  end;
